Applications of bioconvection for tiny particles due to two concentric cylinders when role of Lorentz force is significant

The bioconvection flow of tiny fluid conveying the nanoparticles has been investigated between two concentric cylinders. The contribution of Lorenz force is also focused to inspect the bioconvection thermal transport of tiny particles. The tiny particles are assumed to flow between two concentric cylinders of different radii. The first cylinder remains at rest while flow is induced due to second cylinder which rotates with uniform velocity. Furthermore, the movement of tiny particles follows the principle of thermophoresis and Brownian motion as a part of thermal and mass gradient. Similarly, the gyro-tactic microorganisms swim in the nanofluid as a response to the density gradient and constitute bio-convection. The problem is modeled by using the certain laws. The numerical outcomes are computed by using RKF -45 method. The graphical simulations are performed for flow parameters with specific range like 1≤Re≤5, 1≤Ha≤5, 0.5≤Nt≤2.5, 1≤Nb≤3, 0.2≤Sc≤1.8, 0.2≤Pe≤1.0 and 0.2≤Ω≤1.0. It is observed that the flow velocity decreases with the increase in the Hartmann number that signifies the magnetic field. This outcome indicates that the flow velocity can be controlled externally through the magnetic field. Also, the increase in the Schmidt numbers increases the nanoparticle concentration and the motile density.


Introduction
A nanofluid is a novel class of fluids in which metallic or nonmetallic nanoparticles are scattered over the base fluid. These nanofluids are known for their special heat transfer properties like the presence of nanoparticles pronounced the reflective thermal aspect of base materials [1]. This special property makes these fluids applicable for several engineering applications such as coolants in automobiles, nuclear reactors, solar heaters etc. Some of the special properties of nanoparticles are high specific surface area, higher dispersion stability, reduced particle clogging, and many adjustable properties including thermal conductivity and surface wettability [2]. In this regard, Sheikholeslami [3] studied the impact of porosity and Lorentz force on the heat transfer of nanofluid using Darcy law. Bhamani et al. [4] made contributions for growing heating aspect for turbulent flow of nanofluid inside a pipe. Alrashed et al. [5] modelled a system that describes the flow and the thermal performance of nanofluid comprising of water and MWCNT. Abbas et al. [6] scrutinized the entropy generation in the fully developed flow of nanofluid subjected to velocity slip. Nadeem et al. [7] analyzed the heat transfer characteristics of hybrid nanofluid flowing over a curved surface. The nanomaterials various improved migrated phenomenon with subclass of base fluids was directed in communications Puneeth et al. [8,9]. These continuations on the analysis of heat transfer of single-phase and double phase nanofluid were extended to ternary nanofluid by Manjunatha et al. . Pattern-forming convection movements established in suspensions of paddling microorganisms is known as bio-convection. In liquid suspensions of floating microorganisms, the cellular streaming trend was found wherein fluid flow motions proceed downwards in places where elevated levels of microorganisms develop and swim upwards in regions of low concentration. This sort of pattern is determined by factors including the depth of the suspension, as well as the quantity and mobility of the organisms. Khan  The motion of suspended matter across a fluid in the presence of a temperature gradient is known as thermophoresis. A study to interpret the role of thermophoresis on particle migration and concentration distribution discovered that the concentration distribution gets more non-uniform as the particle size increases. Meanwhile, thermophoresis accentuates non-uniformity in the concentration distribution, with a stronger effect at higher mean concentrations After presenting a comprehensive literature survey, it has been noticed that bioconvection aspect of nanofluid with various flow configurations have been available. However, the bioconvection applications of tiny fluid conveying the nanoparticles between two concentric cylinders different radii is not focused yet. Moreover, the contribution of Lorentz force for bioconvective model is another important task which is addressed in this model. The flow though moving cylinder is interesting topic and some continuations are performed by researchers [41][42][43][44][45][46][47][48][49][50]. This investigation presents the answer of following thermal flow questions: i. Which mathematical model is used to inspect the bioconvection of tiny particles moving between concentric cylinders having different radii?
ii. How heat and mass transfer process fluctuated with interaction of tiny fluid conveying the nanoparticles?
iii. What is contribution of Lorentz force to improve the heating phenomenon?
iv. How thermophoresis and Brownian motion parameters pays role to enhance the thermal process?

Mathematical model
A laminar flow of a Newtonian fluid containing the nanoparticles in the presence of gyro-tactic microorganisms is assumed to flow between two concentric cylinders. Each of these cylinders are of radii R 1 and R 2 such that R 1 >R 2 . The cylinder with radius R 2 is assumed to be stationary and is enclosed in the cylinder of radius R 1 , whose angular momentum is ω 1 . The presence of microorganisms in the system stabilizes the dilute nanoparticle suspension and prevent sedimentation in the system. The dilute nanoparticle suspension is assumed which no fluctuation of movement of microorganisms within the system. This enables the microorganisms to move easily and constitute the macroscopic phenomena termed bio-convection. Thermophoresis and Brownian motion prospective of tiny particles is examined in view of Buongiorno thermal model. Hence, these two slip mechanisms are included in the mathematical model so that the results that are obtained will be close to practicality. Furthermore, the cylinders of radii r 1 and r 2 are maintained at a temperature T 1 and T 2 respectively along with the concentration C 1 and C 2 and the motile density N 1 and N 2 respectively. The flow configuration is shown in Fig 1 using cylindrical coordinates.
The exploration of thermal model for all constraints is presented via following equations [36, 37, 46]: with The non-dimensionless form of model is [36, 37, 46]: The corresponding boundary conditions are The dimensionless parameters involved in this study are defined as

Solution methodology
The transformed Eqs (6)-(9) along with the boundary conditions (10) are remodeled to initial value problem (IVP). This is further simulated with interpretation of well-known RKF -45 method in acquaintance with the shooting method. For the computation purpose, the infinite boundary conditions are considered at η = 10 and the accuracy of the solution is set to the order of 10 −5 . The proper step size is determined in this method for ensuring the validity. Further, these two approximations are compared and if they hold a close agreement with each other than the approximation is considered valid. The whole process is repeated if the approximations obtained do not match each other and the computation is repeated till the desired accuracy is obtained.

Results and discussion
RKF -45 is used to examine the flow of nanofluid between two concentric cylinders in the presence of self-propelled microorganisms. Using appropriate relationships, the equations are non-dimensionalised. The solution was achieved using RKF -45. The outcomes of this study are interpreted through graphs ((2a)-(5d)) and Tables 1 and 2. Fig 2(A)-2(D) shows the impact of Reynolds number (Re) on the nanofluid flow profiles. Following the physical dynamic of Reynolds number, inertial forces grow up when Reynolds number is higher. Such forces show their major impact at the boundary region. As the value of Re goes higher, the viscous force becomes less significant, and the fluid will thus be less viscous and result in faster flow. The up-raise change in velocity due to Re is noted in Fig 2(A) Moreover, the temperature rate of tiny particles increases due to the friction created within the fluid due to its faster flowing rate as shown in Fig 2(B). Similarly, the Fig 2(C) and 2(D) indicated that improvement assigning to Reynolds number enhances the mass concentration and the motile density respectively.
The Hartmann number (Ha) is the ratio of electromagnetic forces to the viscous forces which measures the significance of drag forces resulting from electromagnetic induction and viscous forces. The impact of this parameter is shown in Fig 3(A)-3(D). The velocity of the nanofluid flow is seen to be reduced in Fig 3(A) because of the strong Lorentz force produced. This force acts against the flow and opposes the motion of the fluid by creating friction. Also, the friction thus created will generate additional heat within the nanofluid as a result more impressive temperature field is noted (Fig 3(B)  The impact of thermophoresis (Nt) on θ(r) and ϕ(r) is shown in Fig 4(A) and 4(B) respectively. The increase in the Nt parameter causes the nanoparticles to move from a hotter region to a colder and the nanoparticles dissipate heat into the fluid. As a consequence, the temperature of the nanofluid increases as shown in Fig 4(A). Meanwhile, the movement of nanoparticles becomes faster with the increase in Nt as a result, ϕ(r) at the boundary layer gowsup as reflected in Fig 4(B). Further, the temperature of the nanofluid increases due to the heat generated because of the collision of nanoparticles. Thus increment with increasing trend in referred to Fig 4(C) the zigzag motion of nanoparticles increases with the increase in Nb. During this zigzag motion, the nanoparticles colloid each other and move away from the boundary region as a result the nanoparticle concentration decreases as depicted in Fig 4(D).
The impact of the Schmidt numbers on the F(r) and X(r) is depicted in Fig 5(A) and 5(B) respectively. The Schmidt numbers are inversely proportional to the diffusivities of their corresponding profiles. As a result, as the concentration Schmidt number (Sc) increases, the diffusivity of the nanoparticles reduces, and F(r) at the boundary layer falls, as illustrated in Fig 5  (A). As Sb increases, the motile density diffusivity falls, and the motile density at the border layer drops, as seen in Fig 5(B). Furthermore, as illustrated in Fig 5(C), increasing the Peclet number increases motile density at the border layer. When O upgrade, the microorganism profile declined as shown in Fig 5(D).
The variations of wall shear surface force, local Nusselt number, Sherwood number and motile density number is tabulated in Tables 1 and 2 for changes in the fluid parameters. Table 1 displays the impact of Reynolds number and Hartmann number whereas, Table 2  shows the impact of Schmidt number and slip mechanisms. The increase in the Reynolds number resulted in a decrease in Cf r Nu r , Sh r and Nn r and the same is tabulated in the first row of Table 1. Whereas, it was noticed that the higher values of Ha increased Cf r , Nu r , Sh r and Nn r . Meanwhile, the increase in the Schmidt number increased the Nusselt number whereas it decreased the Sherwood number and motile density number. Furthermore, the increase in the values of Nb, decreased the values of Nu r , Sh r and Nn r , but the higher values of Nt decreased the Nusselt number and enhanced Sherwood number and motile density number.

Conclusions
The applications of Lorentz force for bioconvection transport of tiny fluid conveying tiny particles due to two concentric cylinders is presented. For nanofluid flow, the concentric cylinders attained same center but different radius. The governing equations are made dimensionless ➢ The enhanced flow velocity and heating phenomenon is noted for increasing the Reynolds number.
➢ The increase in the thermophoresis parameter enhances the thermal and mass profiles of the nanofluid.